J Sci Comput (2010) 44: 286–300 DOI 10.1007/s10915-010-9380-3 Radial Basis Function Interpolation on Irregular Domain through Conformal Transplantation Alfa R.H. Heryudono · Tobin A. Driscoll Received: 5 January 2010 / Accepted: 17 May 2010 / Published online: 3 June 2010 © Springer Science+Business Media, LLC 2010 Abstract In this paper, Radial Basis Function (RBF) method for interpolating two dimen- sional functions with localized features defined on irregular domain is presented. RBF points located inside the domain and on its boundary are chosen such that they are the image of conformally mapped points on concentric circles on a unit disk. On the disk, a fast RBF solver to compute RBF coefficients developed by Karageorghis et al. (Appl. Numer. Math. 57(3):304–319, 2007) is used. Approximation values at desired points in the domain can be computed through the process of conformal transplantation. Some numerical experiments are given in a style of a tutorial and MATLAB code that solves RBF coefficients using up to 100,000 RBF points is provided. Keywords Radial basis functions · Schwarz-Christoffel mapping · Meshfree · Interpolation 1 Introduction Methods based on Radial Basis Functions (RBF) have been widely used for scattered data interpolation in higher dimension. In order to form the RBF interpolant, one requires only a set of nodes called centers, which define the basis functions, and its corresponding interpo- lation data values. These minimal requirements carry over to higher dimensions. The basis functions are radially symmetric with respect to its centers and their shapes remain invariant in all dimensions. The centers themselves do not have to be specially distributed in such a way where interpolation data is difficult to obtain. Moreover, the accuracy of approximation provided by the interpolant can be adjusted using parameters known as shape parameters. More in-depth study about RBF methods can be found in [4, 12, 33] and references therein. A.R.H. Heryudono () Department of Mathematics, University of Massachusetts Dartmouth, Dartmouth, MA 02747, USA e-mail: [email protected] T.A. Driscoll Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA e-mail: [email protected] J Sci Comput (2010) 44: 286–300 287 The flexibilities of RBF method described above regarding its meshfree and/or mesh- tolerant nature and the availability of adjustable parameters make the RBF interpolant a viable choice for an approximant for solutions of partial differential equations on irregular geometry [23, 25]. The freedom of placing nodes wherever needed is a great deal if the func- tion to be interpolated has some localized features. In that case, nodes can be denser around high activity areas and coarser in other region. Indeed, study about adaptive process for au- tomatically refining/coarsening RBF centers based on the profile of the functions/solutions is becoming an active research area [10, 21, 22, 29]. On the other hand, those flexibilities combined with several drawbacks of the RBF method can make the process of selecting the best centers’ distributions and choosing the best parameters a frustrating task. Moreover, as the number of centers grows, the method needs to solve Ax = b with relatively large and dense matrix A. Accuracy of RBF ap- proximations is plagued by severe ill-conditioning of the interpolation matrix that makes achieving higher accuracy difficult. This instability behavior leads to the famous classical accuracy and stability trade-off: RBF uncertainty principle by Schaback [30]. Either one goes for a small error and gets a bad sensitivity, or one wants a stable algorithm and has to take a comparably large error. Several techniques have been devised to tackle this issue such as complex contour in- tegration technique [17], RBF-QR [15, 16], and RBF Matrix Decomposition [24]. If one wishes to use an iterative method, ill-conditioning can also create a serious convergence is- sue. In such cases good preconditioners are needed [3, 13]. In addition to instability from linear algebra, node and center locations also play a crucial role. The classical problem of in- terpolation stability is usually measured by Lebesgue constants and manifested through the Runge phenomenon. In one dimensional case, the interpolant generated by Gaussian RBFs with equally-spaced centers can be transformed into a polynomial [27]. Just as in polyno- mial potential theory, as the number of centers grows to infinity, an interpolated function must be analytic in a complex domain larger than the interval of approximation, unless a special node density is used. This density clusters nodes toward the ends of the interval in order to avoid Runge oscillations. The existence and construction of stable node sets in higher dimensions and general geometries, in particular with regards to proper clustering near the boundary, remains a very challenging problem today. In this article, we focus on using the RBF method for interpolating two dimensional functions with some degree of localization defined on irregular domains. The objective of this paper is to combine two techniques, an FFT-based fast RBF solver by Karageorghis et al. [24] and conformal mapping to handle the irregular geometry. Although all functions used in our numerical experiments can be extended outside the domain, we only use centers inside the domain and on the boundary. The irregular domains are currently chosen to be simply connected polygons. RBF centers are not chosen randomly. Instead, they are the images of conformally mapped points on concentric circles on a unit disk. On the disk, a fast RBF solver [24] that takes advantage of block circulant structures of the interpolation matrix is available. Once the RBF interpolant on the disk is obtained, interpolants between the disk and the irregular domain can be communicated through the process of conformal transplantation. The idea of conformal transplantation between two domains is not new. It has been used in Chebyshev pseudospectral methods [14, 20, 32] as well. One of striking applications is the eigenvalue problems on fractal regions [2]. 288 J Sci Comput (2010) 44: 286–300 2 Radial Basis Functions in Two Dimensional Case We briefly describe the method of interpolation by RBF. For further study, one should con- sult the three references we mentioned in the beginning paragraph of the previous section. c c c c R2 Given a set of N distinct centers (x1,y1),...,(xN ,yN ) in , the RBF interpolant takes the form N j s(x,y) = λj φ (x, y), (1) j=1 j c c where φ (x, y) is a radial basis function centered at (xj ,yj ). While there are a large num- ber of known RBFs, the RBFs that are infinitely differentiable and contain a free shape parameter ε have been the most widely used due to (under certain conditions [5, 7, 26, 34]) their high-order (even spectrally) accurate approximations. As an example, the Inverse Mul- tiquadric (IMQ) 1 φj (x, y) = , (2) + 2 2 1 ε rj (x, y) 2 = − c 2 + − c 2 represents this category of RBF where rj (x, y) (x xj ) (y yj ) is the square of c c Euclidean distance from the center (xj ,yj ). The coefficients λ are determined by enforcing the interpolation condition s(xi ,yi ) = F(xi ,yi ) (3) at a set of nodes that typically coincide with the centers, where F(x,y) is the function to be interpolated. Enforcing the interpolation conditions at N centers results in solving a N × N linear system Aλ = F. (4) The matrix A with entries = j c c = aij φ xi ,yi ,i,j 1,...,N (5) is called the interpolation matrix. An interesting connection exists between spectral that are based on polynomials and the non-polynomial RBF approximations. In the limit ε → 0 the RBF interpolant is equivalent to the minimal-degree Lagrange interpolating polynomial [9]. In higher dimensions, the limit may not exist but when it does it is a low degree multivariate polynomial [31]. How fast RBF approximations converge is recently discussed in [26]. 3 RBF Centers on Concentric Circles Suppose we arrange RBF centers in a systematic way on the concentric circles 2(i − 1)π 2β π 2(i − 1)π 2β π x = R cos + j ,y= R sin + j , (6) i,j j n n i,j j n n where i = 1,...,n, j = 1,...,m with R1 <R2 < ···<Rm and 0 ≤ βj ≤ 1. The N × N RBF interpolation matrix A in (4), where N = nm, provides a nice structure in the form of J Sci Comput (2010) 44: 286–300 289 block circulant submatrices ⎡ ⎤ A11 A12 ··· A1m ⎢ ··· ⎥ ⎢A21 A22 A2m ⎥ A = ⎢ . ⎥ . (7) ⎣ . .. ⎦ Am1 Am2 ··· Amm Following the convention in (5), the entries of the n × n circulant submatrix Akl are formed by using the RBFs centered at points on the circle with radius Rl and evaluated at points on the circle with radius Rk . In this case, RBF coefficients, which are the solution of (4), can be efficiently computed by taking advantage of circulant matrix decomposition via Fast Fourier Transform (FFT). Regarding this method, one should consult the paper by Karageorghis et al. [24]. We briefly describe their method. = A circulant matrix C circ(c1,...,cn) has a nice property such that it can be decom- = ∗ = = n (k−1)(j−1) posed as C U DU,whereD diag(d1,...,dn) with dj k=1 ckω ,andU (Fourier matrix) is the conjugate of an n × n matrix ⎡ ⎤ 11 1 ··· 1 ⎢ 2 n−1 ⎥ ⎢1 ωω··· ω ⎥ ∗ 1 ⎢ 2 4 ··· 2(n−1) ⎥ U = √ ⎢1 ω ω ω ⎥ , where ω = e2πi/n. n ⎢. ⎥ ⎣. .. ⎦ 1 ωn−1 ω2(n−1) ··· ω(n−1)(n−1) Hence, the interpolation matrix (7) that contains block circulant structures can be decom- posed as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∗ U 0 ··· 0 D11 D12 ··· D1m U 0 ··· 0 ⎢ ∗ ··· ⎥ ⎢ ··· ⎥ ⎢ ··· ⎥ ⎢ 0 U 0 ⎥ ⎢D21 D22 D2m ⎥ ⎢ 0 U 0 ⎥ A = ⎢ .